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Trinomial Bonsais

In his recent post on convection, Michael Aichinger recapitulated the treatment of convection in a trinomial tree when convection plays a significant role. In such cases, the standard branching would deliver one negative weight, leading to instabilities. Therefor down- and upbranching have to be implemented.

Too large timesteps?
However, due to the explicitness of trinomial trees, there is a second source of instability in trinomials: When timesteps are too large compared to the discretization in the space (i.e. the short rate) dimension, the scheme becomes unstable. Assume we want to valuate a fixed income instrument under a one-factor Hull-White model with a time horizon of 30 yearsm and we would likt o have a grid resolution for the interest rates of 10 basis points. With a typical (absolute) volatility of, say, 1 percent (a reasonable guess ), this leads to a time step of the order of 1 day and that 10000 time steps are needed.

The grid then looks 50 times finer (in both directions) than the following plot.

Only 200 of the necessary 10000 timesteps plotted here.

No. There are much cleverer methods available.

For a more detailed description of the stability conditions, see section 4.5 of the Workout.